INDEX CORRELATIONS. 43 



evaluate the coefficient in the usual way. Now, if in calculating the 

 indices the carpopodite lengths and cephalothorax lengths were put 

 together in pairs quite at random (say the carpopodite length of indi- 

 vidual X was divided by the cephalothorax length of another individual, 

 F, instead of by its own), instead of as they actually occur, there would 

 still be a correlation between these indices and the length of the cepha- 

 lothorax. This correlation we know would be negative, and its amount 

 may be indicated by the heavy line at B. Now, if there were no organic 

 correlation between our two characters, index and absolute dimensions, 

 obviously we should expect that the observed correlation would be that 

 which arises for arithmetic reasons solely. In other words, we should 

 expect observed and spurious to be equal, or, on the diagram, A and B 

 to coincide. But the two are not equal. From the spurious or arithmetic 

 correlation value at B there has been a shift in the direction of the inside 

 dotted arrow, through "no correlation" at zero to a positive correlation 

 measured by the observed coefficient at A. This shift is the result of 

 the organic correlation between the index and the absolute dimension. 

 A study of the values of table 20 in connection with the diagram will, 

 we believe, convince even the non-mathematical reader of the reality of 

 the result that the proportions of the crayfish body are not independent 

 of its absolute size, but that, instead, the two things are correlated together 

 to a definite and significant degree. 



In order to bring out in another way the generality of this result we 

 have resorted to a still different method. We have directly determined the 

 correlation between an index and an absolute dimension which does not 

 enter as one of the factors in the index. In this case there is no spurious 

 correlation; the total observed relationship is organic in origin, just as 

 truly as the correlation between two absolute dimensions is organic. On 

 account of the considerable arithmetical labor involved we have worked 

 out completely only one of these cases, but it will serve to demonstrate 

 the point made. Other examples were carried far enough to show that 

 essentially the same results would be obtained with them, the differences 

 being only in the numerical values of the coefficients. In order to make 

 the tests as fair as possible the characters which were to go to form the 

 indices were chosen at random. The first pair of characters which were 

 drawn were (a) meripodite of leg I, and (6) propodite of leg I. The 

 quotient of a/6 was directly calculated for each individual and entered 

 on the record cards. Then a correlation table was formed between these 



